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Taylor-Couette Flow of an Oldroyd-B Fluid in an Annulus Subject to a Time-dependent Rotation

Received: 31 May 2015     Accepted: 1 June 2015     Published: 15 June 2015
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Abstract

In this paper the velocity field and the adequate shear stress corresponding to the rotational flow of an Oldroyd-B fluid, between two infinite coaxial circular cylinders, are determined by applying the finite Hankel transforms. The motion is produced by the inner cylinder that, at time t = 0+, is subject to a time-dependent rotational shear stress. The solutions that have been obtained are presented under series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for Maxwell, second grade and Newtonian fluids are obtained as limiting case of general solutions. Finally, the influence of the pertinent parameters on the velocity and shear stress of the fluid is analyzed by graphical illustrations.

Published in American Journal of Applied Mathematics (Volume 3, Issue 3-1)

This article belongs to the Special Issue Proceedings of the 1st UMT National Conference on Pure and Applied Mathematics (1st UNCPAM 2015)

DOI 10.11648/j.ajam.s.2015030301.15
Page(s) 25-31
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2015. Published by Science Publishing Group

Keywords

Taylor-Couette Flow, Oldroyd-B Fluid, Velocity Field, Shear Stress

References
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[5] K. R. Rajagopal, R. K. Bhatnagar, Exact solutions for some simple flows of Oldroyd-B fluid, Acta Mech. 113, pp. 233-239 , 1995 .
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[8] T. Hayat, M. Khan, and T. Wang, Non-Newtonian flow between concentric cylinders, Commun. Nonlinear Sci. Numer. Simul. 11, pp. 297-305, 2006.
[9] C. Fetecau, Corina Fetecau, D. Vieru, On some helical flows of Oldroyd-B fluids, Acta Mech. 189, pp. 53-63, 2007.
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[11] R. Ellahi, T. Hayat, T. Javed, S. Asghar, On the analytic solution of nonlinear flow problem involving Oldroyd 8-constant fluid, Math. Comput. Model. 48, pp. 1191-1200, 2008.
[12] Corina Fetecau, C. Fetecau, M. Imran, Axial Couette flow of an Oldroud-B fluid due to a time-dependent shear stress, Math. Reports 11 (61), No. 2, pp. (2009) 145-154.
[13] Corina Fetecau, M. Imran, C. Fetecau, I. Burdujan, Helical flow of an Oldroyd-B fluid due to a circular cylinder subject to time dependent shear stresses, Z. Angew. Math. Phys., 61, pp. 959-969, 2010.
[14] R. Ellahi, T. Hayat, F. M. Mahomed, and A. Zeeshan, Exact solutions for flows of an Oldroyd 8-constant fluid with nonlinear slip conditions, Commun. Nonlinear Sci. Numer. Simul. 15 , pp. 322-330 , 2010.
[15] T. Hayat, S. Najam, M. Sajid, and M. Ayub, On exact solutions of oscillatory flows in a generalized Burger's fluid with slip condition, Z. Naturforsch. 65a No. 4, pp. 381-390, 2010.
[16] Corina Fetecau, M. Imran And C. Fetecau, Taylor-Couette flow of an Oldroyd-B fluid in an annulus due to a time-dependent couple, Zeit. Nat. A 66a, pp. 40-46 , 2011.
[17] N. D. Waters and M. J. King, Unsteady flow of an elastico-viscous liquid, Rheol. Acta 9, pp. 345–-355, 1970.
[18] R. Bandelli, K. R. Rajagopal, Start-up flows of second grade fluids in domains with one finite dimension, Int. J. Non-Linear Mech. 30, pp. 817-839, 1995.
[19] M. E. Erdogan, On unsteady motions of fluids of a second-order fluid, Int. J. Non-Linear Mech. 38, pp. 1045-1051, 2003.
[20] C. Fetecau and K. Kannan, A note on an unsteady flow of an Oldroyd-B fluid, Int. J. Math. Math. Sci. 19, pp. 3185-3194 , 2005.
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[22] M. Kamran , M. Imran, M. Athar , M. A. Imran, On the unsteady rotational flow of fractional Oldroyd-B fluid in cylindrical domains, Meccanica, DOI 10.1007/s11012-011-9467-4.
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Cite This Article
  • APA Style

    M. Imran, Madeeha Tahir, M. A. Imran, A. U. Awan. (2015). Taylor-Couette Flow of an Oldroyd-B Fluid in an Annulus Subject to a Time-dependent Rotation. American Journal of Applied Mathematics, 3(3-1), 25-31. https://doi.org/10.11648/j.ajam.s.2015030301.15

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    ACS Style

    M. Imran; Madeeha Tahir; M. A. Imran; A. U. Awan. Taylor-Couette Flow of an Oldroyd-B Fluid in an Annulus Subject to a Time-dependent Rotation. Am. J. Appl. Math. 2015, 3(3-1), 25-31. doi: 10.11648/j.ajam.s.2015030301.15

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    AMA Style

    M. Imran, Madeeha Tahir, M. A. Imran, A. U. Awan. Taylor-Couette Flow of an Oldroyd-B Fluid in an Annulus Subject to a Time-dependent Rotation. Am J Appl Math. 2015;3(3-1):25-31. doi: 10.11648/j.ajam.s.2015030301.15

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  • @article{10.11648/j.ajam.s.2015030301.15,
      author = {M. Imran and Madeeha Tahir and M. A. Imran and A. U. Awan},
      title = {Taylor-Couette Flow of an Oldroyd-B Fluid in an Annulus Subject to a Time-dependent Rotation},
      journal = {American Journal of Applied Mathematics},
      volume = {3},
      number = {3-1},
      pages = {25-31},
      doi = {10.11648/j.ajam.s.2015030301.15},
      url = {https://doi.org/10.11648/j.ajam.s.2015030301.15},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ajam.s.2015030301.15},
      abstract = {In this paper the velocity field and the adequate shear stress corresponding to the rotational flow of an Oldroyd-B fluid, between two infinite coaxial circular cylinders, are determined by applying the finite Hankel transforms. The motion is produced by the inner cylinder that, at time t = 0+, is subject to a time-dependent rotational shear stress. The solutions that have been obtained are presented under series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for Maxwell, second grade and Newtonian fluids are obtained as limiting case of general solutions. Finally, the influence of the pertinent parameters on the velocity and shear stress of the fluid is analyzed by graphical illustrations.},
     year = {2015}
    }
    

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    T1  - Taylor-Couette Flow of an Oldroyd-B Fluid in an Annulus Subject to a Time-dependent Rotation
    AU  - M. Imran
    AU  - Madeeha Tahir
    AU  - M. A. Imran
    AU  - A. U. Awan
    Y1  - 2015/06/15
    PY  - 2015
    N1  - https://doi.org/10.11648/j.ajam.s.2015030301.15
    DO  - 10.11648/j.ajam.s.2015030301.15
    T2  - American Journal of Applied Mathematics
    JF  - American Journal of Applied Mathematics
    JO  - American Journal of Applied Mathematics
    SP  - 25
    EP  - 31
    PB  - Science Publishing Group
    SN  - 2330-006X
    UR  - https://doi.org/10.11648/j.ajam.s.2015030301.15
    AB  - In this paper the velocity field and the adequate shear stress corresponding to the rotational flow of an Oldroyd-B fluid, between two infinite coaxial circular cylinders, are determined by applying the finite Hankel transforms. The motion is produced by the inner cylinder that, at time t = 0+, is subject to a time-dependent rotational shear stress. The solutions that have been obtained are presented under series form in terms of Bessel functions, satisfy all imposed initial and boundary conditions. Moreover, these solutions satisfy both the governing differential equations and all imposed initial and boundary conditions. The corresponding solutions for Maxwell, second grade and Newtonian fluids are obtained as limiting case of general solutions. Finally, the influence of the pertinent parameters on the velocity and shear stress of the fluid is analyzed by graphical illustrations.
    VL  - 3
    IS  - 3-1
    ER  - 

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Author Information
  • Department of Mathematics, Government College University, Faislabad, Pakistan

  • Department of Mathematics, Government College University, Faislabad, Pakistan

  • School of Science and Technology, Department of Mathematics, University of Management and Technology Lahore, Lahore, Pakistan

  • Department of Mathematics, University of the Punjab, Lahore, Pakistan

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